Worksheet for practicing box and whisker plots with data analysis questions.

Let’s solve this step by step.

We are given 20 data points (mile times in minutes):

8, 7.5, 6, 5.75, 6.25, 6.75, 6.5, 8.75, 9, 8
9.25, 8, 8.25, 8.25, 9.25, 8, 6.75, 7, 4.5, 8.5

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Step 1: Order the data from least to greatest

Let’s list them in order:

4.5,
5.75,
6,
6.25,
6.5,
6.75,
6.75,
7,
7.5,
8,
8,
8,
8,
8.25,
8.25,
8.5,
8.75,
9,
9.25,
9.25

✔ Total = 20 numbers — correct.

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Step 2: Find the Five-Number Summary

The five-number summary is:
- Minimum
- Q1 (First Quartile)
- Median (Q2)
- Q3 (Third Quartile)
- Maximum

→ Minimum = smallest number = 4.5

→ Maximum = largest number = 9.25

→ Median (Q2): Since there are 20 numbers (even), median is average of 10th and 11th values.

Ordered list positions:

1: 4.5
2: 5.75
3: 6
4: 6.25
5: 6.5
6: 6.75
7: 6.75
8: 7
9: 7.5
10: 8 ← 10th
11: 8 ← 11th
12: 8
... etc.

So median = (8 + 8) / 2 = 8

→ Q1 (First Quartile): This is the median of the *first half* of the data (first 10 numbers).

First 10 numbers:

4.5, 5.75, 6, 6.25, 6.5, 6.75, 6.75, 7, 7.5, 8

Again, even number → average of 5th and 6th:

5th = 6.5
6th = 6.75
Q1 = (6.5 + 6.75) / 2 = 6.625

→ Q3 (Third Quartile): Median of the *second half* (last 10 numbers)

Last 10 numbers (positions 11 to 20):

8, 8, 8.25, 8.25, 8.5, 8.75, 9, 9.25, 9.25, 9.25? Wait — let's list carefully:

From position 11 to 20:

11: 8
12: 8
13: 8
14: 8.25
15: 8.25
16: 8.5
17: 8.75
18: 9
19: 9.25
20: 9.25

Wait — actually, after position 10 (which was 8), the next 10 are:

Position 11: 8
12: 8
13: 8
14: 8.25
15: 8.25
16: 8.5
17: 8.75
18: 9
19: 9.25
20: 9.25

Yes.

Now find median of these 10: average of 5th and 6th in this subset.

In this subset:

1st: 8 (pos11)
2nd: 8 (pos12)
3rd: 8 (pos13)
4th: 8.25 (pos14)
5th: 8.25 (pos15)
6th: 8.5 (pos16)
7th: 8.75
8th: 9
9th: 9.25
10th: 9.25

So 5th = 8.25, 6th = 8.5
Q3 = (8.25 + 8.5) / 2 = 8.375

✔ So Five-Number Summary:

- Minimum: 4.5
- Q1: 6.625
- Median: 8
- Q3: 8.375
- Maximum: 9.25

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Step 3: Create a scale for your number line above

We need to cover from 4.5 to 9.25.

A good scale would be from 4 to 10, with marks every 0.5 or 1 unit.

Since we have decimals like .25, .5, .75, maybe use increments of 0.5.

Scale suggestion: Start at 4.0, then 4.5, 5.0, 5.5, ..., up to 9.5 or 10.0.

But since max is 9.25, going to 9.5 is fine.

You can label: 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5

That covers all our data.

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Step 4: Make the box-and-whisker plot

Draw a number line with the scale above.

Mark:

- Left whisker end: min = 4.5
- Left edge of box: Q1 = 6.625
- Line inside box: median = 8
- Right edge of box: Q3 = 8.375
- Right whisker end: max = 9.25

Connect with lines (whiskers) from min to Q1, and Q3 to max. Box between Q1 and Q3.

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Step 5: Shade sections representing 25% each

The box-and-whisker plot divides data into 4 equal parts (quartiles), each 25%.

So:

- From min to Q1: first 25% → shade one color
- Q1 to median: second 25% → another color
- Median to Q3: third 25% → another color
- Q3 to max: fourth 25% → last color

Label each section “25%” as instructed.

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Step 6: Answer questions below

> The first quartile has mile times from 5.25 to 6.25. TRUE or FALSE

Our Q1 is 6.625, and minimum is 4.5. First quartile means the lowest 25% of data — which goes from min (4.5) to Q1 (6.625). But the statement says “from 5.25 to 6.25” — that’s not correct. Also, 5.25 isn’t even in our data! Our actual first quartile range is 4.5 to 6.625.

Also, note: sometimes people confuse “first quartile” as the value Q1, but here it says “has mile times from...”, meaning the range of the first 25%.

So: FALSE.

Correct range for first 25%: 4.5 to 6.625

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> What percentage of the mile times range from 7.25 to 9.375?

Look at our ordered data:

We want how many values are ≥7.25 and ≤9.375.

List again:

4.5,
5.75,
6,
6.25,
6.5,
6.75,
6.75,
7,
7.5 ← starts here (≥7.25)
8,
8,
8,
8,
8.25,
8.25,
8.5,
8.75,
9,
9.25,
9.25 ← all up to here are ≤9.375

So from 7.5 onward: that’s positions 9 to 20 → 12 values.

Total = 20 → 12/20 = 60%

Wait — let’s count:

Positions 9 to 20 inclusive: 20 - 9 + 1 = 12 values.

Yes.

But wait — is 7.25 included? Our data doesn’t have 7.25. The first value ≥7.25 is 7.5 (position 9). And 9.375 — our max is 9.25, so all values up to 9.25 are included.

So yes, 12 out of 20 = 60%

Answer: 60%

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> Calculate the range, mean, and mode.

→ Range = max - min = 9.25 - 4.5 = 4.75

→ Mean: Add all 20 numbers, divide by 20.

Let’s add:

List:

4.5
5.75
6
6.25
6.5
6.75
6.75
7
7.5
8
8
8
8
8.25
8.25
8.5
8.75
9
9.25
9.25

Add step by step:

Group them:

Start with whole numbers and easy decimals.

4.5 + 5.75 = 10.25
+6 = 16.25
+6.25 = 22.5
+6.5 = 29
+6.75 = 35.75
+6.75 = 42.5
+7 = 49.5
+7.5 = 57
+8 = 65
+8 = 73
+8 = 81
+8 = 89
+8.25 = 97.25
+8.25 = 105.5
+8.5 = 114
+8.75 = 122.75
+9 = 131.75
+9.25 = 141
+9.25 = 150.25

Total sum = 150.25

Mean = 150.25 ÷ 20 = 7.5125

→ Mode: Most frequent value.

Count frequencies:

4.5: 1
5.75: 1
6: 1
6.25: 1
6.5: 1
6.75: 2
7: 1
7.5: 1
8: appears 4 times (positions 10,11,12,13) → wait, let’s check:

In ordered list:

After 7.5 (pos9), then:

pos10: 8
pos11: 8
pos12: 8
pos13: 8 → that’s four 8s? Wait no:

Wait original unordered had:

Looking back at original data:

Original 20 values:

Row1: 8, 7.5, 6, 5.75, 6.25, 6.75, 6.5, 8.75, 9, 8 → that’s two 8s already

Row2: 9.25, 8, 8.25, 8.25, 9.25, 8, 6.75, 7, 4.5, 8.5 → three more 8s? Let’s count total 8s:

From row1: 8 (first), 8 (last) → 2
Row2: 8 (second), 8 (sixth), and also 8.5? No — 8.5 is different.

Wait row2: “9.25, 8, 8.25, 8.25, 9.25, 8, 6.75, 7, 4.5, 8.5”

So 8 appears at position 2 and 6 in row2 → that’s two more → total 4 eights?

But in ordered list we had:

At positions 10,11,12,13: all 8s? That would be four.

But let’s recount original data explicitly:

Given data:

First row:
8, 7.5, 6, 5.75, 6.25, 6.75, 6.5, 8.75, 9, 8 → 8 appears twice

Second row:
9.25, 8, 8.25, 8.25, 9.25, 8, 6.75, 7, 4.5, 8.5 → 8 appears twice more (positions 2 and 6)

So total 8s: 2 + 2 = 4

Also, 8.25 appears twice, 9.25 appears twice, 6.75 appears twice.

So mode is 8 (appears 4 times, more than any other).

✔ So:

Range: 4.75
Mean: 7.5125
Mode: 8

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> What is the interquartile range? Explain.

IQR = Q3 - Q1 = 8.375 - 6.625 = 1.75

Explanation: The interquartile range is the spread of the middle 50% of the data. It tells us how far apart the first and third quartiles are. A smaller IQR means the middle half of the data is closer together; larger means more spread out.

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> What do you notice about how many times were between 5.25 and 9.25? Explain.

First, look at our data: min is 4.5, max is 9.25.

Values between 5.25 and 9.25: exclude anything <5.25 or >9.25.

Our data: only 4.5 is below 5.25. All others are ≥5.75, which is >5.25.

And max is 9.25, so nothing above.

So only one value (4.5) is outside [5.25, 9.25].

Thus, 19 out of 20 times are between 5.25 and 9.25.

That’s 95%.

Why? Because 4.5 is an outlier on the low side. The rest are clustered higher.

Note: 5.25 isn't in our data, but we’re asked about the interval.

So answer: Almost all (19 out of 20) mile times fall between 5.25 and 9.25 minutes. Only one time (4.5) is below that range. This shows most students ran between roughly 5.25 and 9.25 minutes, with one very fast runner.

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> What is the purpose of a box and whisker plot?

It shows the spread and center of a dataset using the five-number summary. It helps you see where the middle 50% of data lies (the box), and how spread out the extremes are (whiskers). It’s great for comparing groups or spotting outliers.

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Final Answer:

Five-Number Summary:
- Minimum: 4.5
- Q1: 6.625
- Median: 8
- Q3: 8.375
- Maximum: 9.25

Scale suggestion: Use number line from 4.0 to 9.5, marked every 0.5 units.

Box-and-whisker plot: Draw with left whisker at 4.5, box from 6.625 to 8.375, median line at 8, right whisker at 9.25. Shade each quartile section (min-Q1, Q1-median, median-Q3, Q3-max) with different colors and label each “25%”.

True/False: FALSE — first quartile ranges from 4.5 to 6.625, not 5.25 to 6.25.

Percentage from 7.25 to 9.375: 60%

Range: 4.75
Mean: 7.5125
Mode: 8

Interquartile Range: 1.75 — it measures the spread of the middle 50% of the data.

Times between 5.25 and 9.25: 19 out of 20 (95%) — only 4.5 is below, showing most times are in that range.

Purpose of box-and-whisker plot: To visually display the distribution of data using the five-number summary, highlighting central tendency, spread, and potential outliers.